AREA of RESEARCH

Finite Geometry, Algebraic Geometry, Coding Theory

RECENT PUBLICATIONS

BOOKS

The following books form a 3-volume treatise on the geometry and combinatorics of finite projective spaces:

The first volume, PGOFF, deals with the background, some generalities, and then the line and the plane. The second volume, FPSOTD, considers 3-dimensional space as well as some aspects of 5-dimensional space. The third volume, GGG, considers n-dimensions.

The new book, to appear later this year or early next year, begins with a classical account of curves over an arbitrary field and continues with a detailed account in the case that the field is finite.

PAPERS (For an offprint or preprint, please write to me: jwph@sussex.ac.uk)

    1. Algebraic curves and arcs

    • (with G. Korchmáros) On the embedding of an arc into a conic in a finite plane.
      Finite Fields Appl. 2 (1996), 274-292.
      conic.ps
      A subset of a finite plane of odd order is a k-arc if it consists of k points with at most two on any line of the plane. The largest such set is a conic, that is the set of zeros of an irreducible quadratic form. The question is how big a k -arc has to be to ensure that it is contained in a conic. This paper improves the lower bound. The method involves a detailed study of algebraic curves over a finite field.

    • (with G. Korchmáros) On the number of rational points of an algebraic curve over a finite field.
      Bull. Belg. Math. Soc. Simon Stevin 5 (1998), 313-340.
      num.ps

      This paper continues the study of curves, and obtains an upper bound for the number of rational points of a curve; the bound is linear in the degree of the curve and is also valid for reducible curves with no linear or quadratic component. This bound improves the Hasse-Weil bound for a certain range of the degree. A consequence is a further improvement of the lower bound of the previous paper.

    • (with G. Korchmáros) Arcs and algebraic curves over a finite field.
      Finite Fields Appl. 5 (1999), 393-408.
      even.ps

      This paper shows that for q an even square that the interval between the sizes of the second and third largest complete k-arc is at least the square root of q. This required an improved bound on the number of rational points of a curve.

    • (with A. Cossidente, G. Korchmáros and F. Torres) On plane maximal curves.
      Compositio Math. 121 (2000), 163-181.
      max.ps

      This paper gives a characterization of the curves that achieve the upper bound in the second paper. It is shown that plane curves achieving the Hasse-Weil upper bound and having degree half that of a Hermitian curve are Fermat curves.

    • (with G. Korchmáros) On the number of solutions of an equation over a finite field.
      Bull. London Math. Soc. 33 (2001), 16-24.
      number.ps Abstract: lumin.ps

      This paper extends the plane case of the Stöhr-Voloch theorem on an upper bound for the number of rational points of a curve to obtaining an upper bound for the number of branches centred at a point in the ground field. A lower bound is also obtained.

    • (with M. Giulietti and G. Korchmáros) The Desarguesian plane of order thirteen.
      Finite Geometries, Developments of Mathematics, Kluwer, 2001, 159-170.
      13.ps

      This paper shows how Noether's theorem applied to the algebraic curve associated to an arc in PG(2,q), with q odd, gives both properties of the curve itself as well as properties of the arc. The key case of (q-1)-arcs means that the behaviour of the associated sextic curves needs to be studied. The case of PG(2,13) is examined in detail. There is a geometric bijection between 12-arcs and their duals. The latter lead to optimal sextic curves of genus 10 over GF(13); the former lead to sextics whose set of rational points make them `look like' quartics.

    • (with G. Korchmáros) Caps on Hermitian varieties and maximal curves.
      Adv. Geom., Special Issue (2003), 206-214.
      cap.ps

      This paper considers complete caps on Hermitian varieties and particularly the three-dimensional case. Here, a complete cap is a maximal set of points meeting each line of the Hermitian surface in at most one point. Maximal curves can be embedded in the surface as curves of degree q+ 1, and are examples of complete caps.

    • (with M. Giulietti, G. Korchmáros and F. Torres) Curves covered by the Hermitian curve.
      Finite Fields Appl. 12 (2006), 539-564.
      ghkt1.ps

      This paper describes a family of q/12 maximal non-isomorphic curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point.

    • (with M. Giulietti, G. Korchmáros and F. Torres) A family of curves covered by the Hermitian curve.
      Sémin. Congr., to appear.
      ghkt2.ps

      This paper extends the results of the previous paper to a more general family.

    2. Surveys on constants in finite spaces

    • (with L. Storme) The packing problem in statistics, coding theory and finite geometry.
      J. Statist. Plann. Inference 72 (1998), 355-380.
      pack.ps
    • (with L. Storme) The packing problem in statistics, coding theory and finite geometry: update 2001.
      Finite Geometries, Developments of Mathematics, Kluwer, 2001, 201-246.
      survey.ps

      These are both survey papers, whose main topic is the maximum size of a subset of a projective or vector space over a finite field subject to certain simple intersection conditions with all subspaces of a certain dimension. This includes the best known bounds on arcs, caps and blocking sets.

    • The 1959 Annali di Matematica paper of Beniamino Segre and its legacy.
      Topics in Combinatorics: Geometry, Graph Theory and Designs, Combinatorics 2002, J. Geometry 76 (2003), 82-94.
      alta4.ps

      This paper considers how bounds for the number of rational points on an algebraic curve can be applied to geometric problems.

    3. MDS codes and complete arcs

    • The main conjecture for MDS codes
      Cryptography and Coding. Lecture Notes in Computer Science 1025, Springer, 1995, pp. 44-52.
      main.ps
    • Complete arcs
      Discrete Math. 174 (1997), 177-184.
      compl.ps

    These two papers survey the latest situation on the Main Conjecture for Maximum Distance Separable Codes. The conjecture states that, for q > k-1, the maximum length n of a linear [n,k,n-k+1] code over GF(q) is q+1 unless q is even and k = 3 or q-1, in which case q+2 is the answer.

    4. Projective geometry codes

    • (with D.G. Glynn) On the classification of geometric codes by algebraic functions.
      Des. Codes Cryptogr. 6 (1995), 189-204.
    • (with R. Shaw) Projective geometry codes over prime fields. Finite Fields: Theory, Applications and Algorithms
      Contemporary Mathematics 168, American Mathematical Society, 1994, pp. 151-163.

    These two papers describe the connection of a subset of a finite space with a function on the subspace. The former produces an explanation for a general formula for the dimension of the associated code; the latter examines the calculus of these function spaces in detail and finds a new, single summation formula for the dimension when the field has prime order.

    5. Line geometry

    • (with A. Cossidente and L.Storme) Applications of line geometry, III. The quadric Veronesean and the chords of a twisted cubic
      Australas. J. Combin. 16 (1997), 99-111.

    6. Surveys on algebraic geometry codes

    • Codes on curves and their geometry.
      Rend. Circ. Mat. Palermo Suppl. 51 (1998), 123-137.
    • Curves and configurations in finite spaces.
      Rend. Circ. Mat. Palermo Suppl. 53 (1998), 69-87.

    The first is a survey paper on algebraic geometry codes for a special number of the journal on Recent Progress in Geometry.

    The second is a survey paper on the number of rational points on an algebraic curve and the characterization of maximal curves.

    7. Geometry on a Hermitian surface

    • (with G.L. Ebert) Complete systems of lines on a Hermitian surface over a finite field.
      Des. Codes Cryptogr. 17 (1999), 253-268.

    A non-singular Hermitian surface U3,q in PG(3,q) cannot be partitioned by lines lying on it. So the question of the maximum number of skew lines on it is considered. The case q=9 is considered in detail; it produces a remarkable Kummer configuration.

    8. Arcs and linear codes

    • (with S. Ball) Bounds on (n,r)-arcs and their application to linear codes.
      Rend Mat. Appl., to appear
      krarcs.ps

    This article reviews some of the principal and recently-discovered lower and upper bounds on the maximum size of (n,r)-arcs in PG(2,q), sets of n points with at most r points on a line. Some of the upper bounds are used to improve the Griesmer bound for linear codes in certain cases. Also, a table is included showing the current best upper and lower bounds for q up to 19, and a number of open problems are discussed.
CONFERENCE PROCEEDINGS

  1. (with S.S. Magliveras and M.J. de Resmini) Geometry, Combinatorial Designs and Related Structures. Proceedings of the First Pythagorean Conference, London Math. Soc. Lecture Notes Series, Cambridge University Press 245, 1997.

    These are the proceedings of an advanced research workshop on finite geometries and designs held on the Greek island of Spetses in June 1996. It contains state-of-the-art surveys by leading practitioners and research papers by the top mathematicians in their fields. The invited speakers contributed the following major articles: Cameron considers the current state of finite geometry from a group-theoretical viewpoint; Jungnickel and Schmidt give an authoritative survey of difference sets; Lindner surveys small embeddings of partial cycle systems into Steiner triple systems; Mathon reports successful searches for spreads and packings of designs; Shult describes rank three geometries with simplicial residues; Thas and Van Maldeghem characterize generalized quadrangles satisfying Veblen's Axiom. Additionally, there are articles on new 7-designs, biplanes, various aspects of triple systems, generalized quadrangles with parameters (q2,q), unitals, applications of algebraic geometry in finite geometry, spreads and parallelisms of projective spaces.

  2. (with S.S. Magliveras and M.J. de Resmini) Proceedings of the Second Pythagorean Conference, J. Geometry 67, 2000.

    These are the proceedings of a similar workshop on finite geometries and designs held on the Greek island of Samos in June 1999. For the contents, see the information on the conference on JWPH's home page.

  3. (with A. Blokhuis, D. Jungnickel, J.A. Thas) Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference, Developments in Mathematics, Kluwer, 2001.

    These 21 articles, containing both surveys and original papers, arise from this research conference held in July 2000.

    The main speakers were A.R. Calderbank, P.J. Cameron, C.E. Praeger, B. Schmidt, H. Van Maldeghem.

    The themes covered include diagram geometries, configurations in finite projective spaces, generalized quadrangles, generalized polygons, design theory, difference sets, coding theory, permutation groups, and algorithms.

    For more detailed contents, see the information on the conference on JWPH's home page.

  4. Surveys in Combinatorics, 2001, London Math. Soc. Lecture Note Series 288, Cambridge University Press, 2001.

    These 9 articles are the survey papers contributed by the nine invited speakers at the 18th British Combinatorial Conference, held at the University of Sussex in in July 2001, as well as an obituary. The contributors are J. Sheehan, M. Aigner, I. Anderson, A.R. Calderbank and A.F. Naguib, L.A. Goldberg, B. Mohar, M.S.O. Molloy, J.G. Oxley, J.A. Thas, D.R. Woodall,

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